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Properties

Label 2-1008-7.5-c2-0-17
Degree $2$
Conductor $1008$
Sign $0.291 - 0.956i$
Analytic cond. $27.4660$
Root an. cond. $5.24080$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.85 + 4.53i)5-s + (−1.17 + 6.90i)7-s + (8.81 + 15.2i)11-s − 19.0i·13-s + (14.7 − 8.53i)17-s + (7.74 + 4.46i)19-s + (−4.72 + 8.18i)23-s + (28.6 + 49.6i)25-s + 10.0·29-s + (39.5 − 22.8i)31-s + (−40.5 + 48.8i)35-s + (−26.9 + 46.6i)37-s − 58.9i·41-s − 69.3·43-s + (−41.7 − 24.0i)47-s + ⋯
L(s)  = 1  + (1.57 + 0.907i)5-s + (−0.168 + 0.985i)7-s + (0.801 + 1.38i)11-s − 1.46i·13-s + (0.869 − 0.501i)17-s + (0.407 + 0.235i)19-s + (−0.205 + 0.355i)23-s + (1.14 + 1.98i)25-s + 0.345·29-s + (1.27 − 0.736i)31-s + (−1.15 + 1.39i)35-s + (−0.727 + 1.26i)37-s − 1.43i·41-s − 1.61·43-s + (−0.887 − 0.512i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.291 - 0.956i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.291 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.291 - 0.956i$
Analytic conductor: \(27.4660\)
Root analytic conductor: \(5.24080\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1),\ 0.291 - 0.956i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.757011865\)
\(L(\frac12)\) \(\approx\) \(2.757011865\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (1.17 - 6.90i)T \)
good5 \( 1 + (-7.85 - 4.53i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-8.81 - 15.2i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 19.0iT - 169T^{2} \)
17 \( 1 + (-14.7 + 8.53i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-7.74 - 4.46i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (4.72 - 8.18i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 10.0T + 841T^{2} \)
31 \( 1 + (-39.5 + 22.8i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (26.9 - 46.6i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 58.9iT - 1.68e3T^{2} \)
43 \( 1 + 69.3T + 1.84e3T^{2} \)
47 \( 1 + (41.7 + 24.0i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-21.6 - 37.4i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (0.930 - 0.537i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-45.3 - 26.2i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-24.0 - 41.5i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 53.5T + 5.04e3T^{2} \)
73 \( 1 + (2.16 - 1.25i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-11.2 + 19.5i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 51.2iT - 6.88e3T^{2} \)
89 \( 1 + (43.9 + 25.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 126. iT - 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.06170846653687731631652692532, −9.420928078023117434275590433340, −8.349034882643381593548877950267, −7.23176149434277044503512062761, −6.47087171091422355963504641566, −5.66541684188278478336873579243, −5.04945019568861391947389524093, −3.30803251892427349773662386334, −2.52152740057939772133495923815, −1.51697583352488770380053080603, 0.944534369693899586767846435702, 1.69376027832511673763876565562, 3.26477991405325750925578892799, 4.37494689306761562533020757932, 5.29104222653062787220872981921, 6.33644477515115451855018521560, 6.67536985794516179704411994729, 8.212872022683486845952785647459, 8.842354150947217691201340907565, 9.673910313535724545590375728343

Graph of the $Z$-function along the critical line